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Magazine Chapter 6: Problem 48
Factor completely. $$ a^{2}-a b-12 b^{2} $$
Short Answer
Expert verified
\[ (a + 3b)(a - 4b) \].
Step by step solution
01
Understand the Problem
The exercise involves factoring the quadratic expression completely: \[ a^{2}-a b-12 b^{2} \].
02
Identify Coefficients
The quadratic expression can be identified as being in the form \[ ax^2 + bx + c \], where:\[ a = 1 \]\[ b = -b \]\[ c = -12b^2 \].
03
Find Two Numbers
Find two numbers that multiply to \[ a \times c = 1 \times -12b^2 = -12b^2 \] and add to \[ b = -1b \].
04
Determine Factors
The two numbers are \[ 3b \] and \[ -4b \] since \[ 3b \times (-4b) = -12b^2 \] and \[ 3b + (-4b) = -b \].
05
Rewrite the Middle Term
Rewrite the expression by splitting the middle term using the factors found:\[ a^2 + 3ab - 4b^2 \].
06
Factor by Grouping
Factor by grouping:\[ a^2 + 3ab - 4b^2 \]Rewrite as:\[ a(a + 3b) - 4b(a + 3b) \].
07
Combine Like Terms
Combine like terms to complete the factoring:\[ (a + 3b)(a - 4b) \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
Quadratic equations are mathematical expressions of the form \[ax^2 + bx + c = 0\], where \(a\), \(b\), and \(c\) are constants and \(a e 0\). In the provided exercise, the equation given is \(a^2 - ab - 12b^2\).
The goal when factoring a quadratic expression is to rewrite it as a product of binomials. This helps in solving the equation and understanding its roots.
For example, if you factor \(x^2 - 5x + 6\), you get \( (x - 2)(x - 3) \). Setting this to zero gives solutions \(x = 2\) and \(x = 3\).
Understanding how to transform and manipulate these expressions is crucial in algebra. This will make solving quadratic equations much easier.
The goal when factoring a quadratic expression is to rewrite it as a product of binomials. This helps in solving the equation and understanding its roots.
For example, if you factor \(x^2 - 5x + 6\), you get \( (x - 2)(x - 3) \). Setting this to zero gives solutions \(x = 2\) and \(x = 3\).
Understanding how to transform and manipulate these expressions is crucial in algebra. This will make solving quadratic equations much easier.
Factoring by Grouping
Factoring by grouping is a method used when a quadratic expression is not easily factorable through simple inspection. This method helps break down complex expressions into simpler factors. In the context of the exercise \(a^2 - ab - 12b^2\), we use grouping to interpret and simplify the expression.
Here's how it works:
In the exercise, this approach helps transform \(a^2 - ab - 12b^2\) into \(a(a + 3b) - 4b(a + 3b) \). Notice how we've grouped the terms to factor out common variables in each group.
Here's how it works:
- Find two numbers that multiply to the product of the coefficient of the first term and the last term, and add up to the coefficient of the middle term.
- Rewrite the quadratic expression using these two numbers to split the middle term.
- Group the terms in pairs to factor out common elements.
In the exercise, this approach helps transform \(a^2 - ab - 12b^2\) into \(a(a + 3b) - 4b(a + 3b) \). Notice how we've grouped the terms to factor out common variables in each group.
Splitting Middle Term
Splitting the middle term is a strategy often used within the factoring by grouping method. This technique involves rewriting the middle term in a quadratic expression as the sum of two terms.
Here's how to do it:
For the expression \(a^2 - ab - 12b^2\): We found two numbers (3b and -4b) that multiply to \(-12b^2\) and add up to \(-ab\). Thus, we rewrite the expression as \(a^2 + 3ab - 4b^2\).
This allows us to factor by grouping, ultimately leading us to the factors \((a + 3b)(a - 4b)\).
This method simplifies solving quadratic equations, making it a powerful tool in algebra.
Here's how to do it:
- Identify and multiply the coefficients of the first and last terms of the expression.
- Find two numbers whose product equals the above result and whose sum/difference equals the coefficient of the middle term.
- Rewrite the middle term using these two numbers.
For the expression \(a^2 - ab - 12b^2\): We found two numbers (3b and -4b) that multiply to \(-12b^2\) and add up to \(-ab\). Thus, we rewrite the expression as \(a^2 + 3ab - 4b^2\).
This allows us to factor by grouping, ultimately leading us to the factors \((a + 3b)(a - 4b)\).
This method simplifies solving quadratic equations, making it a powerful tool in algebra.
